Graphs - College Algebra
Card 0 of 380
Find all horizontal and vertical asymptotes in the graph of
.
Find all horizontal and vertical asymptotes in the graph of .
To find vertical asymptotes, factor each quadratic and then simplify.

The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.
Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line
(where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.
To find vertical asymptotes, factor each quadratic and then simplify.
The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.
Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line (where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.
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Where are the vertical asymptotes of this rational function?

Where are the vertical asymptotes of this rational function?
The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of
.
We can use the quadratic formula,
, for a polynomial
. In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:

And we get the two roots,
.
The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of .
We can use the quadratic formula, , for a polynomial
. In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:
And we get the two roots, .
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Which is a vertical asymptote of the graph of the function
?
(a) 
(b) 
Which is a vertical asymptote of the graph of the function
?
(a)
(b)
The vertical asymptote(s) of the graph of a rational function such as
can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator and denominator.
The numerator is a perfect square trinomial and can be factored as such:

The denominator can be factored as the difference of squares:

Rewrite

as

The expression can be reduced by cancelling
in both halves:

Set the denominator equal to 0 and solve:



The only vertical asymptote is therefore the line of the equation
.
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator and denominator.
The numerator is a perfect square trinomial and can be factored as such:
The denominator can be factored as the difference of squares:
Rewrite
as
The expression can be reduced by cancelling in both halves:
Set the denominator equal to 0 and solve:
The only vertical asymptote is therefore the line of the equation .
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Which is a vertical asymptote of the graph of the function
?
(a) 
(b) 
Which is a vertical asymptote of the graph of the function ?
(a)
(b)
The vertical asymptote(s) of the graph of a rational function such as
can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for
:



The graph of the line
is the only vertical asymptote of the graph of
.
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for
:
The graph of the line is the only vertical asymptote of the graph of
.
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Which of the following is a vertical asymptote of the graph of the function
?
(a) 
(b) 
Which of the following is a vertical asymptote of the graph of the function ?
(a)
(b)
The vertical asymptote(s) of the graph of a rational function such as
can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator. It is a quadratic trinomial with lead term
, so look to "reverse-FOIL" it as

We seek two integers whose sum is
and whose product is
; through trial and error, we find
and 2, so

Therefore,
can be rewritten as

Cancelling
, this can be seen to be essentially a polynomial function:

which does not have a vertical asymptote.
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator. It is a quadratic trinomial with lead term , so look to "reverse-FOIL" it as
We seek two integers whose sum is and whose product is
; through trial and error, we find
and 2, so
Therefore, can be rewritten as
Cancelling , this can be seen to be essentially a polynomial function:
which does not have a vertical asymptote.
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Give the equation(s) of the vertical asymptote(s) of the graph of the function
.
Give the equation(s) of the vertical asymptote(s) of the graph of the function
.
The vertical asymptote(s) of the graph of a rational function such as
can be found by evaluating the zeroes of the denominator.
Set the denominator equal to 0 and solve for
:



The only vertical asymptote of the graph is the line of the equation
.
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator.
Set the denominator equal to 0 and solve for :
The only vertical asymptote of the graph is the line of the equation .
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Give the
-coordinate of the
-intercept the graph of the function
.
Give the -coordinate of the
-intercept the graph of the function
.
The
-intercept of the graph of a function is the point at which it intersects the
-axis. The
-coordinate is 0, so the
-coordinate can be found by evaluating
. This is done by substitution, as follows:


The value of the denominator here is 0, so
is an undefined quantity. Consequently, the graph of
has no
-intercept.
The -intercept of the graph of a function is the point at which it intersects the
-axis. The
-coordinate is 0, so the
-coordinate can be found by evaluating
. This is done by substitution, as follows:
The value of the denominator here is 0, so is an undefined quantity. Consequently, the graph of
has no
-intercept.
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Give the
-coordinate(s) of the
-intercept(s) the graph of the function
.
Give the -coordinate(s) of the
-intercept(s) the graph of the function
.
The
-intercept(s) of the graph of
are the point(s) at which it intersects the
-axis. The
-coordinate of each is 0; their
-coordinate(s) are those value(s) of
for which
, so set up, and solve for
, the equation:


A fraction is equal to 0 if and only if the numerator is equal to 0, so set

Factor out
:

By the Zero Product Property, one of the factors must be equal to 0, so either

or

in which case
.
However, setting
in the definition, we see that
, an undefined expression due to the zero denominator. 5 cannot be eliminated similarly. Therefore,
is the only
-intercept.
The -intercept(s) of the graph of
are the point(s) at which it intersects the
-axis. The
-coordinate of each is 0; their
-coordinate(s) are those value(s) of
for which
, so set up, and solve for
, the equation:
A fraction is equal to 0 if and only if the numerator is equal to 0, so set
Factor out :
By the Zero Product Property, one of the factors must be equal to 0, so either
or
in which case
.
However, setting in the definition, we see that
, an undefined expression due to the zero denominator. 5 cannot be eliminated similarly. Therefore,
is the only
-intercept.
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A function
is defined on the domain
according to the above table.
Define a function
. Which of the following values is not in the range of the function
?
A function is defined on the domain
according to the above table.
Define a function . Which of the following values is not in the range of the function
?
This is the composition of two functions. By definition,
. To find the range of
, we need to find the values of this function for each value in the domain of
. Since
, this is equivalent to evaluating
for each value in the range of
, as follows:

Range value: 3

Range value: 5

Range value: 8

Range value: 13

Range value: 21

The range of
on the set of range values of
- and consequently, the range of
- is the set
. Of the five choices, only 45 does not appear in this set; this is the correct choice.
This is the composition of two functions. By definition, . To find the range of
, we need to find the values of this function for each value in the domain of
. Since
, this is equivalent to evaluating
for each value in the range of
, as follows:
Range value: 3
Range value: 5
Range value: 8
Range value: 13
Range value: 21
The range of on the set of range values of
- and consequently, the range of
- is the set
. Of the five choices, only 45 does not appear in this set; this is the correct choice.
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Refer to the above diagram, which shows the graph of a function
.
True or false:
.
Refer to the above diagram, which shows the graph of a function .
True or false: .
The statement is false. Look for the point on the graph of
with
-coordinate
by going right
unit, then moving up and noting the
-value, as follows:

, so the statement is false.
The statement is false. Look for the point on the graph of with
-coordinate
by going right
unit, then moving up and noting the
-value, as follows:
, so the statement is false.
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Define a function
.
Which statement correctly gives
?
Define a function .
Which statement correctly gives ?
The inverse function
of a function
can be found as follows:
Replace
with
:


Switch the positions of
and
:

or

Solve for
. This can be done as follows:
Square both sides:


Add 9 to both sides:


Multiply both sides by
, distributing on the right:



Replace
with
:

The inverse function of a function
can be found as follows:
Replace with
:
Switch the positions of and
:
or
Solve for . This can be done as follows:
Square both sides:
Add 9 to both sides:
Multiply both sides by , distributing on the right:
Replace with
:
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The above diagram shows the graph of function
on the coordinate axes. True or false: The
-intercept of the graph is 
The above diagram shows the graph of function on the coordinate axes. True or false: The
-intercept of the graph is
The
-intercept of the graph of a function is the point at which it intersects the
-axis (the vertical axis). That point is marked on the diagram below:

The point is about one and three-fourths units above the origin, making the coordinates of the
-intercept
.
The -intercept of the graph of a function is the point at which it intersects the
-axis (the vertical axis). That point is marked on the diagram below:
The point is about one and three-fourths units above the origin, making the coordinates of the -intercept
.
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Evaluate: 
Evaluate:
Evaluate the expression
for
, then add the four numbers:






Evaluate the expression for
, then add the four numbers:
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Evaluate: 
Evaluate:
Evaluate the expression
for
, then add the five numbers:







Evaluate the expression for
, then add the five numbers:
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Consider the polynomial
,
where
is a real constant. For
to be a zero of this polynomial, what must
be?
Consider the polynomial
,
where is a real constant. For
to be a zero of this polynomial, what must
be?
By the Factor Theorem,
is a zero of a polynomial
if and only if
. Here,
, so evaluate the polynomial, in terms of
, for
by substituting 2 for
:




Set this equal to 0:


By the Factor Theorem, is a zero of a polynomial
if and only if
. Here,
, so evaluate the polynomial, in terms of
, for
by substituting 2 for
:
Set this equal to 0:
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refers to the floor of
, the greatest integer less than or equal to
.
refers to the ceiling of
, the least integer greater than or equal to
.
Define
and 
Which of the following is equal to
?
refers to the floor of
, the greatest integer less than or equal to
.
refers to the ceiling of
, the least integer greater than or equal to
.
Define and
Which of the following is equal to ?
, so, first, evaluate
by substitution:





, so evaluate
by substitution.




,
the correct response.
, so, first, evaluate
by substitution:
, so evaluate
by substitution.
,
the correct response.
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refers to the floor of
, the greatest integer less than or equal to
.
refers to the ceiling of
, the least integer greater than or equal to
.
Define
and
.
Evaluate 
refers to the floor of
, the greatest integer less than or equal to
.
refers to the ceiling of
, the least integer greater than or equal to
.
Define and
.
Evaluate
, so first, evaluate
using substitution:





, so evaluate
using substitution:




,
the correct response.
, so first, evaluate
using substitution:
, so evaluate
using substitution:
,
the correct response.
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Find all horizontal and vertical asymptotes in the graph of
.
Find all horizontal and vertical asymptotes in the graph of .
To find vertical asymptotes, factor each quadratic and then simplify.

The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.
Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line
(where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.
To find vertical asymptotes, factor each quadratic and then simplify.
The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.
Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line (where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.
Compare your answer with the correct one above
Where are the vertical asymptotes of this rational function?

Where are the vertical asymptotes of this rational function?
The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of
.
We can use the quadratic formula,
, for a polynomial
. In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:

And we get the two roots,
.
The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of .
We can use the quadratic formula, , for a polynomial
. In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:
And we get the two roots, .
Compare your answer with the correct one above
Which is a vertical asymptote of the graph of the function
?
(a) 
(b) 
Which is a vertical asymptote of the graph of the function
?
(a)
(b)
The vertical asymptote(s) of the graph of a rational function such as
can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator and denominator.
The numerator is a perfect square trinomial and can be factored as such:

The denominator can be factored as the difference of squares:

Rewrite

as

The expression can be reduced by cancelling
in both halves:

Set the denominator equal to 0 and solve:



The only vertical asymptote is therefore the line of the equation
.
The vertical asymptote(s) of the graph of a rational function such as can be found by evaluating the zeroes of the denominator after the rational expression is reduced.
First, factor the numerator and denominator.
The numerator is a perfect square trinomial and can be factored as such:
The denominator can be factored as the difference of squares:
Rewrite
as
The expression can be reduced by cancelling in both halves:
Set the denominator equal to 0 and solve:
The only vertical asymptote is therefore the line of the equation .
Compare your answer with the correct one above